Make ScreenAIServiceRouter constructor private.

The service should be only created once per profile, and hence it is
limited to be only created by `ScreenAIServiceRouterFactory`.

AX-Relnotes: n/a
Bug: 330716441
Change-Id: I9620df961e2d56c362e8f25c76866d5e8f60ac72
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5466022


Auto-Submit: Ramin Halavati <rhalavati@chromium.org>
Commit-Queue: Ramin Halavati <rhalavati@chromium.org>
Reviewed-by: Kyungjun Lee <kyungjunlee@google.com>
Cr-Commit-Position: refs/heads/main@{#1289437}